Quadratics In the X-B Plane
By: Lacy Gainey
is know as the standard form of a quadratic equation. When it comes to quadratic equations in secondary mathematics, focus is put on finding roots. In this exploration, we are going to concentrate on patterns of roots for quadratic equations.
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Lets take a look at 
. In this equation a=1 and c=1.
Graph this equation in the x-b plane. So we are letting b=y.
We can see that the graph is a hyperbola with two asymptotes. One of these asymptotes is located at x=0, but the other one is a little harder to locate.
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Lets look at the graph of when c=0.
From the graph, we can see that the other asymptote is located at x = -y.
Is this what you expected?
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Next, graph the equation for different values of c. Try c =1, 3, 5, & 7.
Our graphs still contain a hyperbola and two asymptotes. As c increases, the hyperbola seems to moving farther away from the orgin.
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Graph 
and y=5. What observations do you make about the roots of the original equation?
We can see that the line y=5 intersects the curve twice. We have two negative roots when y > 2 and one negative root when y=2.
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Graph and y= -5. What do you notice?
The line y= -5 also intersects the curve twice. This time we have two positive roots when y< -2 and one positive root when y= -2. Additionally, there are no real roots when -2 < b < 2.
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Consider the case when c = -1, rather than c =1. Graph 
and y = 5.
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Compare the roots of to the roots of when y = 5.
The line y = 5 intersects the curve of two times. However, unlike , has both a positive and negative root at y =5.
The animation below illustrates -5 ≤ y ≤ 5. What observations can you make about the roots.
We can see that when c = -1, there will always be two real roots for [5].
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Does this statement hold for all negative values of c? Lets see.
It appears that it does.
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From this exploration we can conclude:
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All quadratic functions in the x-b plane had zero, one, or two roots.
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When c in the equation is equal to 1.
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has two roots when b > 2 or b < -2.
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has one root when b=2 or b = -2.
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has no roots when -2 < b < 2.
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When c is the equation [1] is negative, [1] will always have two roots.
